I know that in 1952 [Jitsuro Nagura][1] was able to show that there is always a prime between $k$ and $\frac{6k}{5}$ for $k > 24$.

At what point would an improvement on Nagura's result be interesting?  If an approach could show for example that for any $k$, there is a specific value $X$ which could be calculated such that for all $x \ge X$, there is a prime between $kx$ and $(k+1)x$, would this be interesting?

Or, does the Prime Number Theorem provide us enough insight that short of a proof of Legendre's Conjecture, elementary results are not very interesting at this time?

 [1]: https://projecteuclid.org/journals/proceedings-of-the-japan-academy-series-a-mathematical-sciences/volume-28/issue-4/On-the-interval-containing-at-least-one-prime-number/10.3792/pja/1195570997.full